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Treebeard's Stumper
14 December 2001

Christmas Cut-Ups

You can make a perfect five-pointed star from a circle of paper with just one straight cut if you fold it right first. But how? You can cut a holiday pie into two pieces with one straight vertical cut, four pieces with two cuts, or seven pieces with three cuts, if you're careful and don't move the pieces between the cuts. How many pieces can you make with ten cuts? Is there a general formula for any number of cuts? I suppose you can experiment with lines on paper, but it will be much more satisfying to experiment with real holiday pies!

0 cuts, 1 piece. 1 cut, 2 pieces. 2 cuts, 4 pieces. 3 cuts, 7 pieces.
 
I don't know if it's actually possible to draw all the pieces of a pie after ten cuts, but I'll have to try! A better strategy for this stumper is to think about the pattern and understand the effect of making one more cut. This stumper can be extended in many ways:
  • Is it possible to draw the pie after 4-10+ cuts with pleasing symmetry like I started doing above?
  • With three different (non-colinear) cuts I get a maximum of 7 pieces and a minimum of 4 pieces if the cuts don't cross. What is the minimum number of pie pieces for additional cuts? Is every in-between value always possible?
  • What about slicing a solid cheese ball into pieces if the straight cuts can come from any direction? This is very hard to visualize, but the math is similar.
  • Cutting a circle is the same as cutting a square, since both plane shapes are convex. But what about cutting a concave shape like a crescent moon or a star? What about slicing a banana?
  • The big question on my mind is how many equal area/volume pieces are possible with n cuts. In my picture of the pie with three cuts, there are seven unequal pieces. It's easy to get the minimum of six equal-area pieces if all cuts meet in the center like a sliced pizza. Is it also possible to adjust the cuts to give seven equal area (not shape) pieces? Is this general?
  • I started this by swiping a nice picture of a bakery pumpkin pie, but after I reduced it to 16 colors, it looks more like the end of a pine log. Now I realize there's another practical stumper here about how to best cut 2x4s (etc.) from a whole log. I'll save that for a future stumper! I love the way one question unexpectedly leads to another. Good questions take us farther than answers!
3 cuts, 6 equal pieces.

Answer


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Copyright © 2001 by Marc Kummel / mkummel@rain.org